Constrained Sequential Lamination: Nonconvex Optimization and Material Microstructure

Citation

Fago, Matthew Justin (2004) Constrained Sequential Lamination: Nonconvex Optimization and Material Microstructure. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/P1PK-E179. https://resolver.caltech.edu/CaltechETD:etd-05142004-144712

Abstract

A practical algorithm has been developed to construct, through sequential lamination, the partial relaxation of multiwell energy densities such as those characteristic of shape memory alloys. The resulting microstructures are in static and configurational equilibrium, and admit arbitrary deformations. The laminate topology evolves during deformation through branching and pruning operations, while a continuity constraint provides a simple model of metastability and hysteresis. In cases with strict separation of length scales, the method may be integrated into a finite element calculation at the subgrid level. This capability is demonstrated with a calculation of the indentation of a Cu-Al-Ni shape memory alloy by a spherical indenter.

In verification tests the algorithm attained the analytic solution in the computation of three benchmark problems. In the fourth case, the four-well problem (of, e.g., Tartar), results indicate that the method for microstructural evolution imposes an energy barrier for branching, hindering microstructural development in some cases. Although this effect is undesirable for purely mathematical problems, it is reflective of the activation energies and metastabilities present in applications involving natural processes.

The method was further used to model Shield's tension test experiment, with initial calculations generating reasonable transformation strains and microstructures that compared well with the sequential laminates obtained experimentally.

Thesis Files